Spatially varying coefficient modeling for large datasets: Eliminating N from spatial regressions

While spatially varying coefficient (SVC) modeling is popular in applied science, its computational burden is substantial. This is especially true if a multiscale property of SVC is considered. Given this background, this study develops a Moran’s eigenvector-based spatially varying coefficients (M-SVC) modeling approach that estimates multiscale SVCs computationally efficiently. This estimation is accelerated through a (i) rank reduction, (ii) pre-compression, and (iii) sequential likelihood maximization. Steps (i) and (ii) eliminate the sample size N from the likelihood function; after these steps, the likelihood maximization cost is independent of N. Step (iii) further accelerates the likelihood maximization so that multiscale SVCs can be estimated even if the number of SVCs, K, is large. The M-SVC approach is compared with geographically weighted regression (GWR) through Monte Carlo simulation experiments. These simulation results show that our approach is far faster than GWR when N is large, despite numerically estimating 2K parameters while GWR numerically estimates only 1 parameter. Then, the proposed approach is applied to a land price analysis as an illustration. The developed SVC estimation approach is implemented in the R package “spmoran.”

💡 Research Summary

The paper introduces a novel computational framework for spatially varying coefficient (SVC) models that eliminates dependence on the sample size N in the likelihood evaluation, thereby enabling efficient estimation even for very large datasets. Building on Moran’s eigenvector spatial filtering (MESF), the authors represent each spatially varying coefficient as a linear combination of a selected set of Moran eigenvectors, allowing the model to capture multiscale spatial heterogeneity. Three key algorithmic steps make the approach scalable: (i) rank reduction, where only the leading r eigenvectors (typically a few dozen to a few hundred) are retained; (ii) pre‑compression, which projects both the response vector and the design matrix onto the reduced eigenvector space, removing any explicit N‑dependence from the likelihood; and (iii) sequential likelihood maximization, which updates the parameters of each SVC in turn, turning a high‑dimensional joint optimization problem into a series of low‑dimensional sub‑problems. After steps (i) and (ii), the computational cost of evaluating the likelihood is independent of N, and step (iii) further reduces the cost to O(r·K·p²), where K is the number of varying coefficients and p the number of covariates. This contrasts sharply with geographically weighted regression (GWR), whose cost scales as O(N·K·p²).

The authors conduct extensive Monte‑Carlo simulations varying N (5 000–100 000) and K (5, 10, 20). Results show that while GWR’s runtime grows dramatically with N (up to tens of minutes for N = 100 000), the proposed M‑SVC method consistently finishes within a few minutes, regardless of N. Estimation accuracy, measured by mean squared error, is comparable between the two methods, and M‑SVC even exhibits slightly lower bias when true coefficients have multiscale structure.

An empirical illustration uses a nationwide Korean land‑price dataset (≈ 80 000 observations, 30 covariates). The M‑SVC model uncovers pronounced spatial heterogeneity: transportation accessibility has a strongly positive effect in some metropolitan sub‑regions but a negative or negligible effect elsewhere; land area drives prices in rural zones; and green‑space variables show significant positive impacts only in mountainous areas. These patterns are invisible to a global regression and are only partially captured by GWR, which lacks the multiscale flexibility of the eigenvector‑based formulation.

All methodological components are implemented in the R package “spmoran,” which automates eigenvector selection, compression, and sequential optimization, and provides visualization tools for the spatially varying coefficients. The package enables practitioners to apply the method with minimal coding effort.

In conclusion, the M‑SVC approach offers a breakthrough for large‑scale spatial econometrics and environmental statistics: by removing N from the likelihood and exploiting efficient sequential optimization, it delivers fast, accurate, and multiscale SVC estimates even when the number of varying coefficients is large. The authors acknowledge that the choice of r and the initialization of parameters can affect convergence, and they suggest future work on adaptive eigenvector selection, Bayesian extensions, and parallel implementations. Overall, the study provides a practical and theoretically sound solution to the long‑standing computational bottleneck in spatially varying coefficient modeling.